Entropy on Riemann surfaces and the Jacobians of finite covers Curtis T. McMullen 20 June, 2010 Abstract This paper characterizes those pseudo-Anosov mappings whose en- tropy can be detected homologically by taking a limit over finite covers. The proof is via complex-analytic methods. The same methods show the natural map g h, which sends a Riemann surface to the M → A Jacobians of all of its finiteQ covers, is a contraction in most directions. Contents 1 Introduction............................ 1 2 Oddorderzeros.......................... 3 3 Siegelspace............................ 3 4 Teichm¨ullerspace......................... 6 5 Contraction............................ 7 6 Conclusion ............................ 8 A The hyperbolic metric via Jacobians of finite covers . 9 Research supported in part by the NSF. 2000 Mathematics Subject Classification: 32G15, 37E30. Keywords: Entropy, pseudo– Anosov maps, Kobayashi metric, Siegel space, Teichm¨uller space, homology. 1 Introduction Let f : S S be a pseudo-Anosov mapping on a surface of genus g with → n punctures. It is well-known that the topological entropy h(f) is bounded below in terms of the spectral radius of f ∗ : H1(S, C) H1(S, C); we have → log ρ(f ∗) h(f). ≤ If we lift f to a map f : S S on a finite cover of S, then its entropy stays → the same but the spectral radius of the action on homology can increase. e e e We say the entropy of f can be detected homologically if h(f) = sup log ρ(f ∗ : H1(S) H1(S)), → where the supremum is taken over alle finitee covers toe which f lifts. In this paper we will show: Theorem 1.1 The entropy of a pseudo-Anosov mapping f can be detected homologically if and only if the invariant foliations of f have no odd-order singularities in the interior of S. The proof is via complex analysis. Hodge theory provides a natural em- bedding g g from the moduli space of Riemann surfaces into the M → A moduli space of Abelian varieties, sending X to its Jacobian. Any char- acteristic covering map from a surface of genus h to a surface of genus g, branched over n points, provides a similar map g,n h h. (1.1) M → M →A It is known that the hyperbolic metric on a Riemann surface X can be reconstructed using the metrics induced from the Jacobians of its finite covers ([Kaz]; see the Appendix). Similarly, it is natural to ask if the Te- ichm¨uller metric on g,n can be recovered from the Kobayashi metric on M h, by taking the limit over all characteristic covers g,n. We will show such A C a construction is impossible. Theorem 1.2 The natural map g,n h is not an isometry for M → Cg,n A the Kobayashi metric, unless dim g,n = 1Q. M It is an open problem to determine if the Kobayashi and Carath´eodory met- rics on moduli space coincide when dim g,n > 1 (see e.g. [FM, Prob 5.1]). M An equivalent problem is to determine if Teichm¨uller space embeds holo- morphically and isometrically into a (possibly infinite) product of bounded 1 symmetric domains. Theorem 1.2 provides some support for a negative an- swer to this question. Here is a more precise version of Theorem 1.2, stated in terms of the lifted map J g,n h Hh T →T → from Teichm¨uller space to Siegel space determined by a finite cover. Theorem 1.3 Suppose the Teichm¨uller mapping between a pair of distinct points X,Y g,n comes from a quadratic differential with an odd order ∈ T zero. Then sup d(J(X), J(Y )) < d(X,Y ), where the supremum is taken overe alle compatible finite covers of X and Y . Conversely, if the Teichm¨uller map from X to Y has only even order singu- larities, then there is a double cover such that d(J(X), J(Y )) = d(X,Y ) (cf. [Kra]). In particular, the complex geodesics generated by squares of holo- e e morphic 1-forms map isometrically into g. The only directions contracted A by the map g h are those identified by Theorem 1.3. M → A Theorem 1.1 followsQ from Theorem 1.3 by taking X and Y to be points on the Teichm¨uller geodesic stabilized by the mapping-class f. It would be interesting to find a direct topological proof of Theorem 1.1. As a sample application, let β Bn be a pseudo-Anosov braid whose ∈ monodromy map f : S S (on the n-times punctured plane) has an odd → order singularity. Then Theorem 1.1 implies the image of β under the Burau representation satisfies log sup ρ(B(q)) < h(f). |q|=1 Indeed, ρ(B(q)) at any d-th root of unit is bounded by ρ(f ∗) on a Z/d cover S [Mc2]. This improves a result in [BB]. Similar statements hold for other e homological representations of the mapping–class group. Notes and references. For C∞ diffeomorphisms of a compact smooth manifold, one has h(f) log sup ρ(f ∗ Hi(X)) [Ym], and equality holds ≥ i | for holomorphic maps on K¨ahler manifolds [Gr]. The lower bound h(f) ≥ log ρ(f ∗ H1(X)) also holds for homeomorphisms [Mn]. For more on pseudo- | Anosov mappings, see e.g. [FLP], [Bers] and [Th]. A proof that the inclusion of g,n into universal Teichm¨uller space is a T contraction, based on related ideas, appears in [Mc1]. 2 2 Odd order zeros We begin with an analytic result, which describes how well a monomial zk of odd order can be approximated by the square of an analytic function. Theorem 2.1 Let k 1 be odd, and let f(z) be a holomorphic function on ≥ the unit disk ∆ such that f(z) 2 = 1. Then | | R k 2 z √k + 1√k + 3 f(z) Ck = < 1. Z∆ z ≤ k + 2 | | Here the integral is taken with respect to Lebesgue measure on the unit disk. n Proof. Consider the orthonormal basis en(z) = anz , n 0, an = 2 ≥ √n + 1/√π, for the Bergman space Lα(∆) of analytic functions on the disk with f 2 = f(z) 2 < . With respect to this basis, the nonzero entries k k2 | | ∞ in the matrixR of the symmetric bilinear form Z(f, g)= f(z)g(z)zk/ z k are | | given by R k 2√n + 1√k n + 1 Z(en, ek−n)= anak−n z = − Z∆ | | k + 2 · In particular, Z(ei, ei) = 0 for all i (since k is odd), and Z(ei, ej) = 0 for all i, j > k. Note that the ratio above is less than one, by the inequality between the arithmetic and geometric means, and it is maximized when n< k/2 < n + 1. Thus the maximum of Z(f,f) / f 2 over L2 (∆) is achieved when | | k k α f = en + en , n = (k 1)/2, at which point it is given by Ck. +1 − 3 Siegel space In this section we describe the Siegel space of Hodge structures on a surface S, and its Kobayashi metric. Hodge structures. Let S be a closed, smooth, oriented surface of genus g. Then H1(S) = H1(S, C) carries a natural involution C(α) = α fixing H1(S, R), and a natural Hermitian form √ 1 α, β = − α β h i 2 ZS ∧ 3 of signature (g, g). A Hodge structure on H1(S) is given by an orthogonal splitting H1(S)= V 1,0 V 0,1 ⊕ such that V 1,0 is positive-definite and V 0,1 = C(V 1,0). We have a natural norm on V 1,0 given by α 2 = α, α . k k h i The set of all possible Hodge structures forms the Siegel space H(S). To describe this complex symmetric space in more detail, fix a splitting H1(S) = W 1,0 W 0,1. Then for any other Hodge structure V 1,0 V 0,1, ⊕ ⊕ there is a unique operator Z : W 1,0 W 0,1 → such that V 1,0 = (I + Z)(W 1,0). This means V 1,0 coincides with the graph of Z in W 1,0 W 0,1. ⊕ The operator Z is determined uniquely by the associated bilinear form Z(α, β)= α, CZ(β) h i on W 1,0, and the condition that V 1,0 V 0,1 is a Hodge structure translates ⊕ into the conditions: Z(α, β)= Z(β, α) and Z(α, α) < 1 if α = 1. (3.1) | | k k Since the second inequality above is an open condition, the tangent space at the base point p W 1,0 W 0,1 is given by ∼ ⊕ 1,0 1,0 TpH(S)= symmetric bilinear maps Z : W W C . { × → } Comparison maps. Any Hodge structure on H1(S) determines an iso- morphism 1,0 1 R V ∼= H (S, ) (3.2) sending α to Re(α) = (α + C(α))/2. Thus H1(S, R) inherits a norm and a complex structure from V 1,0. Put differently, (3.2) gives a marking of V 1,0 by H1(S, R). By com- posing one marking with the inverse of another, we obtain the real-linear comparison map T = (I + Z)(I + CZ)−1 : W 1,0 V 1,0 (3.3) → between any pair of Hodge structures. It is characterized by Re(α) = Re(T (α)). 4 Symmetric matrices. The classical Siegel domain is given by Hg = Z Mg(C) : Zij = Zji and I ZZ 0 .